In the Multiple Regression Model with Two Regressors, the Formula

Essay

In the multiple regression model with two regressors, the formula for the slope of the first explanatory variable is $$\hat { \beta } _ { 1 } = \frac { \sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \sum _ { i = 1 } ^ { n } y _ { i } x _ { 2 i } \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } } { \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \left( \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } \right) ^ { 2 } }$$ (small letters refer to deviations from means as in $z _ { i } = Z _ { i } - \bar { Z }$ ).
An alternative way to derive the OLS estimator is given through the following three step procedure.
Step 1: regress Y on a constant and $X _ { 2 }$ , and calculate the residual (Res1).
Step 2: regress $X _ { 1 }$ on a constant and $X _ { 2 }$ , and calculate the residual (Res2).
Step 3: regress Res1 on a constant and Res2.
Prove that the slope of the regression in Step 3 is identical to the above formula.